Question 1

Compute the gradient of the function f(x, y) =  sin y +  cos x.
A) (  cos y - 2y cos x) i + (2x sin y +  sin x) j
B) (2x sin y +  sin x) i + (  cos y - 2y cos x) j
C) (2x sin y -  sin x) i + (  cos y + 2y cos x) j
D) (  cos y + 2y cos x) i + (2x sin y -  sin x) j
E) (2x sin y) i + (2y cos x) j

Accepted Answer

C

Question 2

Find grad f(1, 0, -1) if f(x, y, z) = xy + yz.
A) i
B) j
C) 0
D) k
E) i + j + k

Accepted Answer

C

Question 3

If f(x, y, z) =  z + cos(z  ), find  f.
A) 2zx sin(z  ) i + 2yz j + (  +  sin(z  )) k
B) 2x sin(z  ) i + 2yz j + (  -  sin(z  )) k
C) -2x sin(z  ) i + 2yz j + (  -  sin(z  )) k
D) -2zx sin(z  ) i + 2yz j + (  -  sin(z  )) k
E) -2zx cos(z  ) i + 2yz j + (  -  cos(z  )) k

Accepted Answer

D

Question 4

Compute div F for F = (2x + yz) i + (  +  ) j + (x sin(z) +  ) k.
A) 2 + 2y +  + cos(z) + 3 
B) 2 + 2y + z  - x cos(z) + 3 
C) 2 + 2y + z  + x cos(z) + 3 
D) 2 + 2y + x cos(z)
E) 2 + 2y +  - cos(z) + 3

Accepted Answer

The answer of Compute div F for F = (2x...

Question 5

Compute curl F for F = (x - z) i + (y - x) j + (z - y) k.
A) - i + j - k
B) i + j + k
C) - i + j + k
D) i - j
E) 0

Accepted Answer

The curl of a vector field F = (P, Q, R) is given by ∇ × F = (∂R/∂y - ∂Q/∂z) i + (∂P/∂z - ∂R/∂x) j + (∂Q/∂x - ∂P/∂y) k. For F = (x - z, y - x, z - y), we have P = x - z, Q = y - x, R = z - y. Calculating each component: ∂R/∂y - ∂Q/∂z = -1 - 0 = -1, ∂P/∂z - ∂R/∂x = -1 - 0 = -1, ∂Q/∂x - ∂P/∂y = -1 - 0 = -1. Thus, curl F = -i + j - k.

Question 6

Define the curl of a vector field F.
A) F × 
B)  F
C)  × F
D)  . F
E)  F

Accepted Answer

The answer of Define the curl of a vector field...

Question 7

Let  be a scalar field and F be a vector field, both assumed to be sufficiently smooth. Which of the following expressions is meaningless?
A) $\textbf{    curl (grad      }$ )
B) $\textbf{      div (curl F)   }$
C) $\textbf{      grad (div F)   }$
D) $\textbf{ div (grad         }$ )
E) $\textbf{      curl (divF)   }$

Accepted Answer

The answer of Let  be a scalar field and...

Question 8

Let = arctan(x) - arctan(z) and  =  . Find a simplified expression for $\textbf{         grad}$( ) × $\textbf{         grad}$( ) .
A)  j
B) 0 (zero vector field)
C)  j
D) 0 (zero scalar field)
E) -  j

Accepted Answer

The answer of Let = arctan(x) - arctan(z) and ...

Question 9

Compute $\textbf{       div F  }$ for$\textbf{     F     }$ =  sin 2x, cos 2y, tan 2z  .
A) 2cos 2x + 2sin 2y +  z
B) -cos 2x + sin 2y +  z
C) 2cos 2x - 2sin 2y +  z
D) cos 2x + sin 2y +  z
E) 2cos 2x - 2sin 2y + 2sec z

Accepted Answer

The answer of Compute \(\textbf{     ...

Question 10

Find .F if F (x, y, z) =  xy  ,  yz, -xyz  .
A) 2y  +  yz - xy
B) y  + 2xyz - xy
C) y  +  z - xy
D) y  +  z + 2 xy
E) 2y  +  z + xy

Accepted Answer

The answer of Find .F if F (x, y, z)...

Question 11

Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.
A) 2y + 2z + 2x
B) 3y + 2z + x
C) y + z + x
D) 2y -2 z + 2x
E) y + 2z + 3x

Accepted Answer

The divergence of a vector field $ \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} $ is given by $ \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} $. For $ \mathbf{F} = (xy + xz)\mathbf{i} + (yz + yx)\mathbf{j} + (zx + zy)\mathbf{k} $, we compute the partial derivatives: $ \frac{\partial P}{\partial x} = y + z $, $ \frac{\partial Q}{\partial y} = z + x $, and $ \frac{\partial R}{\partial z} = x + y $. Adding these gives $ 2y + 2z + 2x $.

Question 12

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x² + y² - 6 and the sphere x² + y² + z² = 26 at the point (-3, 1, 4) on both surfaces.

Accepted Answer

The answer of Find the acute angle (to the nearest...

Question 13

Calculate the divergence of the vector field F(x, y, z) = (  - xz) i + (z  -  ) j - xy  k.
A)  - z - zx  - 2  - y 
B)  + zx  - 2  + 4xy 
C)  - z + zx  - 2  - 4xy 
D)  - z + zx  - 4xy 
E)  + zx  + 2  + 4xy

Accepted Answer

The answer of Calculate the divergence of the vector field...

Question 14

Calculate the curl of the vector field V = x sin y i + cos y j + xy k.
A) x i + y j - x cos y k
B) x i - y j + x cos y k
C) x i - y j - x cos y k
D) x i + y j + x cos y k
E) -x i + y j + y cos y k

Accepted Answer

The curl of a vector field V = P i + Q j + R k is given by ∇ × V = (∂R/∂y - ∂Q/∂z) i + (∂P/∂z - ∂R/∂x) j + (∂Q/∂x - ∂P/∂y) k. Applying this to V = x sin y i + cos y j + xy k, we get the result x i - y j - x cos y k.

Question 15

Calculate the divergence of the vector field F =  y i +  x j + xyz k.
A) 5xy
B) 4xy + yz
C) 6xy
D) 2xy + 2yz + xz
E) 4xy + xz

Accepted Answer

The answer of Calculate the divergence of the vector field...

Question 16

. F = F .  for any sufficiently smooth vector field F.

Accepted Answer

The answer of  . F = F . ...

Question 17

Let w be a function of x, y, and z having continuous second partial derivatives.Calculate curl grad w in terms of those partials.
A)  +  + 
B)  i +  j +  k
C)  i +  j +  k
D) 0
E)  +  +

Accepted Answer

The answer of Let w be a function of x,...

Question 18

Let F = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k be a vector field in 3-space whose components f, g, and h have continuous second partial derivatives. Calculate div curl F in terms of those partials.
A)  +  + 
B) -  -  - 
C)  +  + 
D) 0
E)  -  +

Accepted Answer

The answer of Let F = f(x, y, z) i...

Question 19

The divergence of a vector field F is defined by
A)  F
B)  . F
C)  F
D)  . (F)
E)  × F

Accepted Answer

The answer of The divergence of a vector field F...

Question 20

The curl of a vector field F is defined by
A)  .(F)
B)  × F
C) 
D)  F
E)  . F

Accepted Answer

The answer of The curl of a vector field F...

Quiz 17: Vector Calculus

Compute the gradient of the function f(x, y) = sin y + cos x.

Find grad f(1, 0, -1) if f(x, y, z) = xy + yz.

If f(x, y, z) = z + cos(z ) , find f.

Compute div F for F = (2x + yz) i + ( + ) j + (x sin(z) + ) k.

Compute curl F for F = (x - z) i + (y - x) j + (z - y) k.

Define the curl of a vector field F.

Find .F if F (x, y, z) = xy , yz, -xyz .

Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x² + y² - 6 and the sphere x² + y² + z² = 26 at the point (-3, 1, 4) on both surfaces.

Calculate the divergence of the vector field F(x, y, z) = ( - xz) i + (z - ) j - xy k.

Calculate the curl of the vector field V = x sin y i + cos y j + xy k.

Calculate the divergence of the vector field F = y i + x j + xyz k.

. F = F . 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 for any sufficiently smooth vector field F.

Let w be a function of x, y, and z having continuous second partial derivatives.Calculate curl grad w in terms of those partials.

Let F = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k be a vector field in 3-space whose components f, g, and h have continuous second partial derivatives. Calculate div curl F in terms of those partials.

The divergence of a vector field F is defined by

The curl of a vector field F is defined by

Quiz 17: Vector Calculus

Compute the gradient of the function f(x, y) = sin y + cos x.

Find grad f(1, 0, -1) if f(x, y, z) = xy + yz.

If f(x, y, z) = z + cos(z ) , find f.

Compute div F for F = (2x + yz) i + ( + ) j + (x sin(z) + ) k.

Compute curl F for F = (x - z) i + (y - x) j + (z - y) k.

Define the curl of a vector field F.

Let be a scalar field and F be a vector field, both assumed to be sufficiently smooth. Which of the following expressions is meaningless?

Let = arctan(x) - arctan(z) and = . Find a simplified expression for grad\textbf{ grad} grad (11ee7bac_4a3a_9aaa_ae82_759e3f104991_TB9661_11 ) × grad\textbf{ grad} grad (11ee7bac_77f1_7c2b_ae82_019616e1397c_TB9661_11 ) .

Compute div F \textbf{ div F } div F for F \textbf{ F } F = sin 2x, cos 2y, tan 2z .

Find .F if F (x, y, z) = xy , yz, -xyz .

Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x2 + y2 - 6 and the sphere x2 + y2 + z2 = 26 at the point (-3, 1, 4) on both surfaces.

Calculate the divergence of the vector field F(x, y, z) = ( - xz) i + (z - ) j - xy k.

Calculate the curl of the vector field V = x sin y i + cos y j + xy k.

Calculate the divergence of the vector field F = y i + x j + xyz k.

. F = F . 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 for any sufficiently smooth vector field F.

Let w be a function of x, y, and z having continuous second partial derivatives.Calculate curl grad w in terms of those partials.

Let F = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k be a vector field in 3-space whose components f, g, and h have continuous second partial derivatives. Calculate div curl F in terms of those partials.

The divergence of a vector field F is defined by

The curl of a vector field F is defined by

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x² + y² - 6 and the sphere x² + y² + z² = 26 at the point (-3, 1, 4) on both surfaces.